\subsection{Stability Analysis}
Before getting results of both the Forward Centered Time Scheme and the Crank-Nicolson Scheme we have to do a stability analysis to see when we can use both the schemes.\\
According to the Von Neumann method of stability analysis we have 
\[
V_x^{\tau}=V(x\Delta X, \tau \Delta \tau) + \varepsilon_x^{\tau}
\]
with $V_x^{\tau}$ the computed solution from the Finite Difference scheme, $V(x\Delta X, \tau \Delta \tau)$ the exact solution and $\varepsilon_x^{\tau}$ the error at time level $\tau$ and mesh point $x$. The idea of this method is to find a condition that bounds the error $\varepsilon_x^{\tau}$ as one advances in time.\\
\noindent
If we introduce $\varepsilon_x^{\tau}=A^{\tau}(k)\exp(Jkx\Delta X)$ with $J=\sqrt{-1}$ the imaginary number, then for stability it is required that
\[
|G|=\left|\frac{A_x^{\tau +1}}{A_x^{\tau}}\right|\leq 1 
\]
$\forall kx\Delta X$.\\
\newline
\noindent
For the FTCS it can be derived \textit{(Finite-Difference Methods for Option Pricing - R. Strijkers, D. Z. Wang, G. Cakir, J. Zagoel)} that 
\begin{eqnarray}\label{eq:GFCTS}
G=1-\Delta\tau\sigma^2\frac{2\sin^2\frac{\beta}{2}}{(\Delta x)^2}-\Delta\tau r+\Delta\tau (r-\frac{1}{2}\sigma^2)\frac{J \sin\beta}{\Delta x}
\end{eqnarray}
so this scheme is stable if and only if the right hand side of equation \ref{eq:GFCTS} is less or equal to 1.\\
Whereas it is proven in the same paper that the CNS is always stable.
